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Beyond the Efron–Buchta Identities: Distributional Results for Poisson Polytopes

Abstract

Let \(\varPi \) be a random polytope defined as the convex hull of the points of a Poisson point process. Identities involving the moment-generating function of the measure of \(\varPi \), the number of vertices of \(\varPi \), and the number of non-vertices of \(\varPi \) are proven. Equivalently, identities for higher moments of the mentioned random variables are given. This generalizes analogous identities for functionals of convex hulls of i.i.d points by Efron and Buchta.

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References

  1. Affentranger, F.: The expected volume of a random polytope in a ball. J. Microsc. 151, 277–287 (1988)

    Article  Google Scholar 

  2. Bárány, I.: Random polytopes in smooth convex bodies. Mathematika 39, 81–92 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bárány, I., Reitzner, M.: Poisson polytopes. Ann. Probab. 38, 1507–1531 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bárány, I., Reitzner, M.: On the variance of random polytopes. Adv. Math. 225, 1986–2001 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Buchta, C.: Stochastische Approximation konvexer Polygone. Z. Wahrsch. Verw. Geb. 67, 283–304 (1984)

  6. Buchta, C.: Zufallspolygone in konvexen Vielecken. J. Reine Angew. Math. 347, 212–220 (1984)

    MATH  MathSciNet  Google Scholar 

  7. Buchta, C.: An identity relating moments of functionals of convex hulls. Discrete Comput. Geom. 33, 125–142 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Buchta, C., Müller, J.: Random polytopes in a ball. J. Appl. Probab. 21, 753–762 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  9. Buchta, C., Reitzner, M.: Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. Probab. Theory Relat. Fields 108, 385–415 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Buchta, C., Reitzner, M.: The convex hull of random points in a tetrahedron: solution of Blaschke’s problem and more general results. J. Reine Angew. Math. 536, 1–29 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Calka, P., Yukich, J.E.: Variance asymptotics for random polytopes in smooth convex bodies. Probab. Theory Relat. Fields 152, 435–463 (2014)

    Article  MathSciNet  Google Scholar 

  12. Doetsch, G.: Handbuch der Laplace-Transformation II. Birkhäuser, Basel, Stuttgart (1955)

  13. Efron, B.: The convex hull of a random set of points. Biometrika 52, 331–343 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hug, D.: Random polytopes. In: Spodarev, E. (ed.) Stochastic Geometry, Spatial Statistics and Random Fields. Lecture Notes in Mathematics, vol. 2068, pp. 205–238, Springer, Heidelberg (2013)

  15. Hug, D., Munsonius, G.O., Reitzner, M.: Asymptotic mean values of Gaussian polytopes. Beitr. Algebra Geom. 45, 531–548 (2004)

    MATH  MathSciNet  Google Scholar 

  16. Kingman, J.F.C.: Random secants of a convex body. J. Appl. Probab. 6, 660–672 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  17. Reitzner, M.: Random polytopes and the Efron–Stein jackknife inequality. Ann. Probab. 31, 2136–2166 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Reitzner, M.: Random polytopes. In: Kendall, W., Molchanov, I. (eds.) New Perspectives in Stochastic Geometry, pp. 45–76. Oxford University Press, Oxford (2010)

    Google Scholar 

  19. Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gewählten Punkten. Z. Wahrsch. Verw. Geb. 2, 75–84 (1963)

    Article  MATH  Google Scholar 

  20. Rényi, A., Sulanke, R.: Über die konvexe Hülle von \(n\) zufällig gewählten Punkten II. Z. Wahrsch. Verw. Geb. 3, 138–147 (1964)

    Article  MATH  Google Scholar 

  21. Schneider, R., Weil, W.: Stochastic and Integral Geometry. Probability and Its Applications (New York). Springer, Berlin (2008)

    Google Scholar 

  22. Zinani, A.: The expected volume of a tetrahedron whose vertices are chosen at random in the interior of a cube. Monatsh. Math. 139, 341–348 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

M. Beermann was supported in part by the FWF project P 22388-N13, ‘Minkowski valuations and geometric inequalities’. We are grateful to an anonymous referee for careful reading of the manuscript and numerous helpful suggestions.

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Correspondence to Matthias Reitzner.

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Beermann, M., Reitzner, M. Beyond the Efron–Buchta Identities: Distributional Results for Poisson Polytopes. Discrete Comput Geom 53, 226–244 (2015). https://doi.org/10.1007/s00454-014-9649-7

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  • DOI: https://doi.org/10.1007/s00454-014-9649-7

Keywords

  • Poisson polytope
  • Random polytope
  • Generating function
  • Efron’s identity

Mathematics Subject Classification

  • Primary 60D05
  • Secondary 60G55
  • 52A22